Abstract
In PG(2,q) it is well known that if k is close to q, then any k-arc is contained in a conic. The internal nuclei of a point set form an arc. In this article it is proved that for q odd the above bound on the number of points could be lowered to \(\tfrac{{q + 1}}{2}\) (or even less), if the arc is obtained as the set of internal nuclei of some point set of proper size. Using this result the internal nuclei of point sets of size q + 1 will be studied in higher dimensional spaces, and an application will be presented to so-called threshold schemes.
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References
A. Beutelspacher, Applications of finite geometry to cryptography, CISM Courses and Lectures No. 313, Springer-Verlag Wien, New York (1990) pp. 161–186.
A. Beutelspacher and F. Wettl, On 2-level secret sharing, Designs, Codes and Cryptography, Vol. 3 (1993) pp. 127–134.
A. Bichara and G. Korchmáros, Note on (q +2)-sets in a Galois plane of order q, Annals of Discrete Math., Vol. 14 (1982) pp. 117–122.
B. Segre, Introduction to Galois geometries, (J.W. P. Hirschfeld, ed.), Atti Accad. Naz. Lincei Memorie, Vol. 8 (1967) pp. 133–236.
J. W. P. Hirschfeld and L. Storme, The packing problem is statistics, coding theory and finite projective spaces, J. Statist. Planning Infer., Vol. 72 (1998) pp. 355–380.
L. Storme and T. Szőnyi, Intersection of arcs and normal rational curves in spaces of odd characteristic, Finite Geometry and Combinatorics, (F. De Clerck et al., eds.), Cambridge University Press (1993) pp. 359–378.
T. Szőnyi, k-Sets in PG(2, q) having a large set of internal nuclei, in Combinatorics'88, Volume 2 (A. Barlotti et al., eds.), Mediterranean Press, Rende (1991) pp. 449–458.
J. A. Thas, Complete arcs and algebraic curves in PG(2, q), Journal of Algebra,Vol. 106 (1987) pp. 451–464.
J. A. Thas, Normal rational curves and k-arcs in Galois spaces, Rend. Mat. Vol. 1 (1968) pp. 331–334.
F. Wettl, On the nuclei of a pointset of a finite projective plane, Journal of Geometry, Vol. 30 (1987) pp. 157–163.
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Kovács, I. On the Internal Nuclei of Sets in PG(n,q), q is Odd. Designs, Codes and Cryptography 24, 37–42 (2001). https://doi.org/10.1023/A:1011269128640
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DOI: https://doi.org/10.1023/A:1011269128640